Sequences play a great role in mathematical communication. In mathematical notation, we use sequence ellipsis (. . . ) to denote "obvious" sequences like 1, 2, . . . , 7, and in conceptualizations sequence constructors like (i 2+1) i∈N. Furthermore, sequences have a prominent role as argument sequences of flexary functions. While the former cases can adequately be represented and reasoned about as domain objects in Open- Math and MathML, argument sequences are at the language level, and can only (...) be represented, but not reasoned with, since the necessary (sequence-schematic) axioms cannot be represented in the language. (shrink)

Throughout the twentieth century, the automation of formal logics in computers has created unprecedented potential for practical applications of logic—most prominently the mechanical verification of mathematics and software. But the high cost of these applications makes them infeasible but for a few flagship projects, and even those are negligible compared to the ever-rising needs for verification. One of the biggest challenges in the future of logic will be to enable applications at much larger scales and simultaneously at much lower costs. (...) This will require a far more efficient allocation of resources. Wherever possible, theoretical and practical results must be formulated generically so that they can be instantiated to arbitrary logics; this will allow reusing results in the face of today’s multitude of application-oriented and therefore diverging logical systems. Moreover, the software engineering problems concerning automation support must be decoupled from the theoretical problems of designing logics and calculi; this will allow researchers outside or at the fringe of logic to contribute scalable logic-independent tools. Anticipating these needs, the author has developed the Mmt framework. It offers a modern approach towards defining, analyzing, implementing, and applying logics that focuses on modular design and logic-independent results. This paper summarizes the ideas behind and the results about Mmt. It focuses on showing how Mmt. provides a theoretical and practical framework for the future of logic. (shrink)